# Meaning of “that” in “holomorphic function in the sector S that is continuous”

I have encountered a confusing sentence in a math textbook:

Suppose F is a holomorphic function in the sector S that is continuous on the closure of S.

What does that mean in the above sentence? Does it mean the function F?

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Suppose F is a holomorphic function in the sector S that is continuous on the closure of S

can be interpreted in two ways, because that can refer to the sector or the holomorphic function in the sector. Of course, from the context we know that it really refers to the latter.

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+1 Nice Jasper. – Babak S. May 4 '13 at 7:56
We being those who know that functions may be continuous (globally or locally) whereas sectors are not spoken of as being continuous. – Edwin Ashworth May 4 '13 at 11:02

You'll have to ask the author.

Grammatically, however, that is a restrictive relative pronoun that refers to "the sector S", and the relative clause means that sector S is continuous on the closure of sector S.

Knowing nothing about math, I can't tell you what this means, but that interpretation makes no semantic sense.

Because this makes no sense to me, I'd say that that is supposed to refer to "the holomorphic function F", as it would in a properly written sentence (but still a stylistically awkward kind of "read my mind, please, because I can't be bothered to express myself clearly" sentence):

Suppose F is a holomorphic function in the sector S and is continuous on the closure of S.

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My goodness! The phantom downvoter strikes again! Or else it's payback time. :-) – user21497 May 4 '13 at 5:46
About the hypothesis of payback time: I never downvote. (I have one recorded downvote on my user page, but that's the result of pushing a wrong button a long time ago) – Georges Elencwajg May 4 '13 at 7:30
@GeorgesE: You're not the only one who doesn't like me. I irritate a lot of people here & elsewhere. But a spirited defensiveness is always a pleasure to observe. Millennia of evolution but guilt still motivates the mammalian brain. :-) – user21497 May 4 '13 at 7:35
From my long-ago (and poorly understood) pure mathematics, I believe that your interpretation is correct (the sector is not continuous on the closure of itself) but your and solution may make the problem more confusing. I suspect that the best referent for that is the general holomorphic function not the specific instance F. – Fortiter May 4 '13 at 8:53
@Fortriter: Thank you for pointing this out. As I said, I know zilch about math, so I was just guessing. Those who know something about it will be able to correct my guess if it's wrong. I'll edit my answer if someone can tell me for certain how best to say it. I suppose I'll have to add something like "and the general holomorphic function"? I don't know. I edit physics, chemistry, & computer papers, but never math papers. I never got beyond basic algebra. OTOH, I invite any mathematician out there to edit that sentence so that it's correct. I don't want to give false or bad information. – user21497 May 4 '13 at 9:07